Noise-induced enhancement of network reciprocity in social dilemmas

The network reciprocity is an important dynamic rule fostering the emergence of cooperation among selfish individuals. This was reported firstly in the seminal work of Nowak and May, where individuals were arranged on the regular lattice network, and played the prisoner’s dilemma game (PDG). In the standard PDG, one often assumes that the players have perfect rationality. However, in reality, we human are far from rational agents, as we often make mistakes, and behave irrationally. Accordingly, in this work, we introduce the element of noise into the measurement of fitness, which is determined by the parameter α controlling the degree of noise. The considered noise-induced mechanism remarkably promotes the behavior of cooperation, which may be conducive to interpret the emergence of cooperation within the population.     

Private Games are too Dangerous

Given the difficulty of observing interpersonal relations as they develop within an organization, I use iterated prisoner&2018;s dilemma games to simulate their development. The goal is to understand how trust could develop as a function of private games, that is, as a function of interaction sequences between two people independent of their relationships with other people. My baseline is Axelrod&2018;s results with TIT for TAT showing that cooperation can emerge as the dominant form of interaction even in a society of selfish individuals without central authority. I replicate Axelrod&2018;s results, then show that the results only occur in a rare social context&2014;maximum density networks. Where people form less dense networks by withdrawing from unproductive relationships, as is typical in organizations, the competitive advantage shifts from TIT for TAT to abusive strategies. A devious PUSHY strategy wins in moderate to high density networks. A blatantly HOSTILE strategy wins in less dense networks. Abusive players do well in sparse networks because their abuse is lucrative in the initial exchanges of a relationship&2014;before the other person knows to withdraw. Wise players avoiding the abusive players leaves the abusive players free to concentrate on naive players (con men thrive in big cities). The implication is that what keeps abusive players at bay are friends and acquaintances warning managers away from people known to exploit their colleagues. I reinforce the point with illustrative survey data to conclude that private games are not only too dangerous, but also too rare and too slow to be the foundation for trust within organizations. The results are an evidential call for the sociological intuition that trust and distrust cannot be understood independent of the network context in which they are produced.     

Cooperation in an evolutionary prisoner’s dilemma game with probabilistic strategies

In this work, we investigate an evolutionary prisoner’s dilemma game in structured populations with probabilistic strategies instead of the pure strategies of cooperation and defection. We explore the model in details by considering different strategy update rules and different population structures. We find that the distribution of probabilistic strategies patterns is dependent on both the interaction structures and the updating rules. We also find that, when an individual updates her strategy by increasing or decreasing her probabilistic strategy a certain amount towards that of her opponent, there exists an optimal increment of the probabilistic strategy at which the cooperator frequency reaches its maximum.     

Role of update dynamics in the collective cooperation on the spatial snowdrift games: Beyond unconditional imitation and replicator dynamics

In this paper, we investigate the role of update or imitation rules in the spatial snowdrift game on regular lattices. Three different update rules, including unconditional imitation (UI), replicator dynamics (RD) and the Moran process, are utilized to update the strategies of focal players during the game process in the spatial snowdrift on the lattice. We observe that the aggregate cooperation level between players is largely elevated by using the Moran process in the spatial snowdrift game, when compared to the UI or replicator dynamics. Meanwhile, we carefully explore the dynamical evolution of frequency of cooperators and the cluster formation pattern for these three update rules. Moreover, it is also shown that the evolutionary behavior under the Moran update is independent of and insensitive to the randomly initial configurations of cooperators and defectors. The current results clearly indicate that the introduction of moderate randomness in the strategy update will highly promote the maintenance and persistence of cooperation among selfish individuals, which will be greatly instrumental to deeply understand the evolution of cooperation within many natural, biological and social systems.     

Can remembering history from predecessor promote cooperation in the next generation?

People usually think that helping the next generation to remember history can promote cooperation in dilemma games. We show that is not always the case when agents have memory. Agents play with each neighbor by game history and strategies (such as TFT and WSLS), and the next generation inherits good strategies from the predecessor. We analyze the system’s cooperation ratio by comparing the 2 sources of history at the beginning of each generation: (a) inherited from the predecessor; (b) randomly initialized with different cooperation ratio. We find that with unconditional imitation update rule, agents who remember history get lower cooperation ratio than those who randomly initialize the history; while with replicator rule, higher initial cooperation ratio promotes higher final cooperation. We also do additional experiments to investigate the R, ST, P reciprocity and strategies distribution of the systems.     

Cooperation by indirect revelation through strategic behavior

The paper deals with a one-shot prisoners' dilemma when the players have an option to go to court but cannot verify their testimonies. To solve the problem a second stage is added to a game. At the first stage the players are involved in the prisoners' dilemma and at the second stage they play another game in which their actions are verifiable. In such a setup the information about the actions chosen at the prisoners' dilemma stage can be revealed through strategic behavior of the players during second stage. A mechanism for such revelation in the extended game is described. It provides an existence of a unique sequential equilibrium, which may be obtained by an iterative elimination of dominated strategies and has a number of desirable properties.     

Considering individual satisfaction levels enhances cooperation in a spatial prisoner’s dilemma game

Based on the observance in human society, the satisfaction level of an individual as a result of an obtained payoff depends on personal tendency to some extent; we establish a new model for spatial prisoner’s dilemma games. We describe individual satisfaction as a stochastically deviated value around each of the four payoffs stipulated by a payoff matrix, which is maintained throughout the life of a certain agent. When strategy updating, an agent who refers to his own satisfaction level cannot see neighbors’ satisfaction levels but can only observe neighbors’ accumulated payoffs. By varying the update rule and underlying topology, we perform numerical simulations that reveal cooperation is significantly enhanced by this change. We argue that this enhancement of cooperation is analogous to a stochastic resonance effect, like the payoff noise effects Perc (2006).     

Strategy changing penalty promotes cooperation in spatial prisoner’s dilemma game

Many classical studies suggest that punishment is a useful way to promote cooperation in the well-mixed public goods game, whereas relative evidence in the research of spatial prisoner’s dilemma game is absent. To address this issue, we introduce a mechanism of strategy changing penalty, combining memory and penalty during the update process, into spatial prisoner’s dilemma game. We find that increasing penalty rate or memory length is able to promote the evolution of cooperation monotonously. Compared with traditional version, recorded penalty could facilitate cooperation better. Moreover, through examining the process of evolution, we provide an interpretation for this promotion phenomenon, namely, the effect of promotion can be warranted by an evolution resonance of standard deviation of fitness coefficient. Finally, we validate our results by studying the impact of uncertainty within strategy adoptions on the evolution of cooperation. We hope that our work may shed light on the understanding of the cooperative behavior in the society.     

Structural heterogeneity mediates the effect of community structure on cooperation

Emergence of cooperation in evolutionary prisoner's dilemma game strongly depends on the topology of underlying interaction network. We explore this dependence using community networks with different levels of structural heterogeneity, which are generated by a tunable upper‐bound on the total number of links that any vertex can have. We study the effect of community structure on cooperation by analyzing a finite population analogue of the evolutionary replicator dynamics. We find that structural heterogeneity mediates the effect of community structure on cooperation. In the community networks with low level of structural heterogeneity, community structure has negative effect on cooperation. However, the positive effect of community structure on cooperation appears and enhances with increasing structural heterogeneity. Our work may be helpful for understanding the complexity of cooperative behaviors in social networks. © 2011 Wiley Periodicals, Inc. Complexity, 2012     

Games with randomly disturbed payoffs: A new rationale for mixed-strategy equilibrium points

Equilibrium points in mixed strategies seem to be unstable, because any player can deviate without penalty from his equilibrium strategy even if he expects all other players to stick to theirs. This paper proposes a model under which most mixed-strategy equilibrium points have full stability. It is argued that for any gameΓ the players' uncertainty about the other players' exact payoffs can be modeled as a disturbed gameΓ ^*, i.e., as a game with small random fluctuations in the payoffs. Any equilibrium point inΓ, whether it is in pure or in mixed strategies, can “almost always” be obtained as a limit of a pure-strategy equilibrium point in the corresponding disturbed gameΓ ^* when all disturbances go to zero. Accordingly, mixed-strategy equilibrium points are stable — even though the players may make no deliberate effort to use their pure strategies with the probability weights prescribed by their mixed equilibrium strategies — because the random fluctuations in their payoffs willmake them use their pure strategies approximately with the prescribed probabilities.     

The emergence of cooperation in tie strength models

In this paper, we propose a tie strength model to explain the emergence of cooperation in spatial prisoner's dilemma games, assuming that cooperators preferentially allocate their investments to friends with strong ties. Two types of prisoner's dilemma models are examined in this study: the traditional two-strategy model considering only cooperators and defectors; the expanded three-strategy model consisting cooperators, defectors and extortioners. The results show that tie strength model contributes to the promotion of cooperation in both types of prisoner's dilemma games. However, we point out that the influence of the investment preference is quite different in the two prisoner's dilemma game settings. In the two-strategy prisoner's dilemma game, only small preference contributes to the promotion of cooperation. Once this preference exceeds a critical value, cooperation will be prohibited. We explain this phenomenon by arguing that extremely strong investment preference undermines the ability of cooperative clusters to resist defectors. Moreover, we extend the analysis into the three-strategy case and discover that the catalytic effect of extortioners can eliminate this first up and then down trend in the two-strategy model. The equilibrium fraction of cooperators is always positively correlated to the level of investment preference in three-strategy models.     

Enhanced cooperation in prisoner’s dilemma with aspiration

We develop an aspiration-based dynamic model which leads to enhanced cooperation in the Prisoner’s Dilemma game played by the continuous population of agents. The main idea is to limit the aggregate information available to the agents. The model–a system of three nonlinear differential equations–describes the evolution of the aspiration levels of players who use different strategies, and the evolution of the mean frequency of the cooperative strategy in the system of players. The stationary (partial) cooperation level is calculated explicitly. We demonstrate that, contrary to the similar model with only one global aspiration level, the stationary cooperation level can be greater than half.     

Finitely repeated prisoners’ dilemma experiments without a commonly known end

Using a symmetric two-player prisoners’ dilemma as base game, each player receives a signal for the number of rounds to be played with the same partner. One of these signals is the true number of rounds R while the other is R − 5. Thus both players know that the game has a finite end. They both know that the opponent knows this, but the finite end is not commonly known. As a consequence, both mutual defection and mutual cooperation until the second last round are subgame perfect equilibrium outcomes. We find experimental evidence that many players do in fact cooperate beyond their individual signal round.     

Impact of neighborhood separation on the spatial reciprocity in the prisoner’s dilemma game

The evolutionary game theory is a very powerful tool to understand the collective cooperation behavior in many real-world systems. In the spatial game model, the payoff is often first obtained within a specific neighborhood (i.e., interaction neighborhood) and then the focal player imitates or learns the behavior of a randomly selected one inside another neighborhood which is named after the learning neighborhood. However, most studies often assume that the interaction neighborhood is identical with the learning neighborhood. Beyond this assumption, we present a spatial prisoner’s dilemma game model to discuss the impact of separation between interaction neighborhood and learning neighborhood on the cooperative behaviors among players on the square lattice. Extensive numerical simulations demonstrate that separating the interaction neighborhood from the learning neighborhood can dramatically affect the density of cooperators (ρ_C) in the population at the stationary state. In particular, compared to the standard case, we find that the medium-sized learning (interaction) neighborhood allows the cooperators to thrive and substantially favors the evolution of cooperation and ρ_C can be greatly elevated when the interaction (learning) neighborhood is fixed, that is, too little or much information is not beneficial for players to make the contributions for the collective cooperation. Current results are conducive to further analyzing and understanding the emergence of cooperation in many natural, economic and social systems.     

不完全信息研发合作的形成及参与人行为选择因素

本通过囚徒困境重复博弈模型的思想研究合作研发中参与人合作策略形成的条件，考察了不完全信息研发合作中参与人合作行为的基础，并在此基础上讨论了研发合作中合作预期收益、不合作激励和合作机会成本对参与人研发合作行为选择的影响。     

Norms of Cooperation in a Game of Partnership

We study a model of partnership with costly commitment in an evolutionary game-theoretic framework. We introduce various types of cooperative players with different degrees of strategic sophistication. We find that a necessary condition for cooperation at the social equilibrium is that the probability of a high opportunity value falls below a certain critical threshold. Such condition, however, is not sufficient. If players are too naively cooperative, the permanence of cooperation at the equilibrium is put at risk because naive cooperation entails playing inefficient responses to other equilibrium strategies. As a consequence, naive cooperation will be present at the equilibrium selected by the social dynamics only under somewhat special conditions. If cooperators are more flexible, in that they remain in the partnership only if mated to a first stage cooperator and only in lack of favorable outside opportunities, cooperation is almost granted provided that it is Pareto efficient (perverse lock in cases under unfavorable initial conditions are possible). Finally, the selection of Pareto superior cooperation with certainty may be granted by a certain norm-oriented type of strategically sophisticated behavior, that we call Rawlsian altruism.     

Noncooperative farsighted stable set in an <Emphasis Type="Italic">n</Emphasis>-player prisoners’ dilemma

We examine an n-player prisoners’ dilemma game in which only individual deviations are allowed, while coalitional deviations (even non-binding ones) are not, and every player is assumed to be sufficiently farsighted to understand not only the direct outcome of his own deviation but also the ultimate outcome resulting from a chain of subsequent deviations by other players. We show that there exists a unique, noncooperative farsighted stable set (NFSS) and that it supports at least one (partially and/or fully) cooperative outcome, which is individually rational and Pareto-efficient. We provide a sufficient condition for full cooperation. Further, we discuss the relationship between NFSS and other “stable set” concepts such as the (myopic) von Neumann–Morgenstern stable set, Harsanyi (1974)’s strictly stable set, Chwe (1994)’s largest consistent set, and the cooperative farsighted stable set examined by Suzuki and Muto (2005). The author is very grateful to Professor Eiichi Miyagawa, the editor and the associate editor of this journal for their insightful comments and suggestions. He also acknowledges the financial support of Japan Society for the Promotion of Science [Grant-in-Aid for Scientific Research (C), No. 18530175].     

Two Game Models for Cooperation with Implicit Noncooperation

We propose two flexible game models to represent and analyze cases that cannot be modeled by current game models. One is called sharing creditability game (SCG) and the other is called bottomline game (BLG). The new models transform cooperative games into new games that incorporate auxiliary information (noncooperative in nature) usually neglected in previous theories. The new games will be solved only by traditional noncooperative game theory. When the new solutions are applied to the original games, the solutions can reflect the auxiliary information in addition to the original objectives of the decision makers or players. Generally, the new solutions are different from the cooperative and the noncooperative solutions of the original games. Existing transferable utility (TU) games and noncooperative games will coincide with special cases of the two new game models. Using SCG and BLG, the prisoner’s dilemma can be reformulated and a richer set of decisions can be considered for the players. The two new game models have potential applications in military and socioeconomic situations.This research was partly funded by the College Engineering, Ohio State University.     

Program equilibrium—a program reasoning approach

The concept of program equilibrium, introduced by Howard (Theory and Decision 24(3):203–213, 1988) and further formalised by Tennenholtz (Game Econ Behav 49:363–373, 2004), represents one of the most ingenious and potentially far-reaching applications of ideas from computer science in game theory to date. The basic idea is that a player in a game selects a strategy by entering a program, whose behaviour may be conditioned on the programs submitted by other players. Thus, for example, in the prisoner’s dilemma, a player can enter a program that says “If his program is the same as mine, then I cooperate, otherwise I defect”. It can easily be shown that if such programs are permitted, then rational cooperation is possible even in the one-shot prisoner’s dilemma. In the original proposal of Tennenholtz, comparison between programs was limited to syntactic comparison of program texts. While this approach has some considerable advantages (not the least being computational and semantic simplicity), it also has some important limitations. In this paper, we investigate an approach to program equilibrium in which richer conditions are allowed, based on model checking—one of the most successful approaches to reasoning about programs. We introduce a decision-tree model of strategies, which may be conditioned on strategies of others. We then formulate and investigate a notion of “outcome” for our setting, and investigate the complexity of reasoning about outcomes. We focus on coherent outcomes: outcomes in which every decision by every player is justified by the conditions in his program. We identify a condition under which there exist a unique coherent outcome. We also compare our notion of (coherent) outcome with that of (supported) semantics known from logic programming. We illustrate our approach with many examples.     

Reciprocal cooperation in the Prisoner's Dilemma repeated with random horizon

Summary A standard assumption on the Iterated Prisoner's Dilemma is that the probability of further interaction is constant. Then, cooperation via reciprocity can be guaranteed if and only if this probability is sufficiently high. In this paper we analyze a more realistic model: a discrete random variable, not necessarily geometric, governs the number of interactions. We obtain necessary conditions and sufficient conditions for reciprocal cooperation based on the use of TFT strategy by both players. The obtained conditions are particularized to classes of distributions with monotonous failure rate and to some relevant probability models.